MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1988 Poland - Second Round
1988 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
Hide problems
sum with vectors, convex polyhedron
Given is a convex polyhedron with
k
k
k
faces
S
1
,
…
,
S
k
S_1, \ldots, S_k
S
1
,
…
,
S
k
. Let us denote the vector of length 1 perpendicular to the wall
S
i
S_i
S
i
(
i
=
1
,
…
,
k
i = 1, \ldots, k
i
=
1
,
…
,
k
) directed outside the given polyhedron by
n
i
→
\overrightarrow{n_i}
n
i
, and the surface area of this wall by
P
i
P_i
P
i
. Prove that
∑
i
=
1
k
P
i
⋅
n
i
→
=
0
→
.
\sum_{i=1}^k P_i \cdot \overrightarrow{n_i} = \overrightarrow{0}.
i
=
1
∑
k
P
i
⋅
n
i
=
0
.
3
1
Hide problems
min |BL|^2 + |CM|^2 + |AN|^2
Inside the acute-angled triangle
A
B
C
ABC
A
BC
we consider the point
P
P
P
and its projections
L
,
M
,
N
L, M, N
L
,
M
,
N
to the sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
, respectively. Determine the point
P
P
P
for which the sum
∣
B
L
∣
2
+
∣
C
M
∣
2
+
∣
A
N
∣
2
|BL|^2 + |CM|^2 + |AN|^2
∣
B
L
∣
2
+
∣
CM
∣
2
+
∣
A
N
∣
2
is the smallest.
5
1
Hide problems
any rectange coved by 25 circles of r=2, can be covered by 100 circles of R=1?
Decide whether any rectangle that can be covered by 25 circles of radius 2 can also be covered by 100 circles of radius 1.
4
1
Hide problems
n^{2n} - n^{n+2} + n^n - 1 is divisible by (n - 1 )^3
Prove that for every natural number
n
n
n
, the number
n
2
n
−
n
n
+
2
+
n
n
−
1
n^{2n} - n^{n+2} + n^n - 1
n
2
n
−
n
n
+
2
+
n
n
−
1
is divisible by
(
n
−
1
)
3
(n - 1 )^3
(
n
−
1
)
3
.
2
1
Hide problems
sum x_i > sum y_i if prod x_i >= prod y_i
Given real numbers
x
i
x_i
x
i
,
y
i
y_i
y
i
(
i
=
1
,
2
,
…
,
n
i = 1, 2, \ldots, n
i
=
1
,
2
,
…
,
n
) such that
x
1
≥
x
2
≥
…
≥
x
n
≥
0
,
y
1
>
y
2
>
…
>
y
n
≥
0
,
\qquad x_1 \geq x_2 \geq \ldots \geq x_n \geq 0, \ \ y_1 > y_2 > \ldots > y_n \geq 0,
x
1
≥
x
2
≥
…
≥
x
n
≥
0
,
y
1
>
y
2
>
…
>
y
n
≥
0
,
and
∏
i
=
1
k
x
i
≥
∏
i
=
1
k
y
i
,
for
k
=
1
,
2
,
…
,
n
.
\prod_{i=1}^k x_i \geq \prod_{i=1}^k y_i, \ \ \text{ for } \ \ k=1,2,\ldots, n.
i
=
1
∏
k
x
i
≥
i
=
1
∏
k
y
i
,
for
k
=
1
,
2
,
…
,
n
.
Prove that
∑
i
=
1
n
x
i
>
∑
i
=
1
n
y
i
.
\sum_{i=1}^n x_i > \sum_{i=1}^n y_i.
i
=
1
∑
n
x
i
>
i
=
1
∑
n
y
i
.
1
1
Hide problems
f(x^{n}) is divisible by x-1
Let
f
(
x
)
f(x)
f
(
x
)
be a polynomial,
n
n
n
- a natural number. Prove that if
f
(
x
n
)
f(x^{n})
f
(
x
n
)
is divisible by
x
−
1
x-1
x
−
1
, then it is also divisible by
x
n
−
1
+
x
n
−
2
+
…
+
x
+
x^{n-1} + x^{n-2} + \ldots + x +
x
n
−
1
+
x
n
−
2
+
…
+
x
+
1.